Chp-01-Atoms, Mole Concept, Atomic Structure (e)_doc

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Atoms Molecules, Chemical Arithmetic and Atomic Structure

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Significant figures.

The total number of digits in a number including the last digit whose value is uncertain is called thenumber of significant figures e.g. 12.4267 ±0.0001 has six significant figures.

Rules for determining the number of significant figures

(1) All non-zero digits as well as the zeros between the non-zero digits are significant e.g. 5004 has foursignificant figures.

(2) Zeros to the left of the first non-zero digit in a number are not significant e.g. 0.0045 has twosignificant figures.

(3) If a number ends in zeros but these zeros are to the right of the decimal point, then these zeros aresignificant e.g. 2.500 has four significant figures.

(4) If a number ends in zeros but these zeros are not to the right of a decimal point, these zeros may ormay not be significant e.g. 10500 written as 1.05×104 has three significant figures but written as 1.050×104 hasfour significant figures and so on.

(5) The result of on addition or subtraction should be reported to the same number of decimal places as

that of the term with least number of decimal places e.g. sum 4.525+2.3+6.24=13.063 will be reported as 13.1.(6) The result of multiplication or division should be reported to the same number of significant figures as

is possessed by the least precise term e.g. 4.327×2.8=12.1156 will be reported as 12.

(7) If calculation involves a number of steps, the result should contain the same number of significantfigures as that of the least precise number involved, other than the exact numbers.

Rounding off non significant figures

It implies as follows

(1) If the digit just next to the last digit to be retained is less than 5, the last digit is taken as such and allother digits to its right are dropped e.g. 1.234=1.23.

(2) If the last digit is greater than 5, the digit to be retained is increased by 1 and all other digits on itsright are dropped e.g. 1.236=1.24.

(3) If the digit just next to the last digit to be retained is equal to 5, the last significant figure is leftunchanged if it is even and is increased by 1 if it is odd e.g. 1.235=1.24, 1.225=1.22.

S.l. Units.

Seven basic SI units for physical quantities are as follows

Length Mass Time Temperature ElectricCurrent

LuminousIntensity

Amount of substance

metre(m)

Kilogram(kg)

Second (s) Kelvin ( K ) Ampere ( A) Candela (Cd ) Mole (mol )

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Some derived units : Volume = 3m , Velocity = 1−ms , Acceleration 2−= ms , Force= =−2mskg newton ( N )

Pressure = 21 sec−−mkg = pascal ( Pa), Energy = 22 − smkg = joule ( J ), Density = 3−mkg .

Some prefixes used for subsidiary units : These are either multiples or fractions of the base units

e.g. deci = ,10 1− centi = ,10 2− milli = ,10 3− micro = ,10 6− nano = ,10 9− pico = ,10 12− femto = ,10 15−

(fermi = 1310− cm = ),10 15m− deca = ,10 1− hecta = 210

− , kilo = 310 , Mega = 610 , giga = 109, tera = 1210 , peta =1510 , exa = 1810

Some useful conversion factors : 1Å= 1010− m, 1 nm= 910

− m, 1 pm= 1210 − m, 1 litre = ,110 33 dmm =− 1 atm

= 760 mmor torr = 101, 325 paor 2− Nm , 1 bar 510 Pa, 1 calorie = 4.184 J , 1 electron volt (eV ) = 1.6022 × 1910− J .

Atom and Molecules.

Atom : It is the smallest particle of an element that takes place in a chemical reaction. It may or may not be capable of free existence.

Molecule : It is the smallest particle of an element or a compound that is capable of free existence.

(i) The term molecule was given by Avogadro. (ii) The term element was given by Robert Boyle.

Laws of chemical combinations.Based on the quantitative results of chemical reactions dealing with the weight –weight and weight-

volume relationships, few laws were formulated which are -

Law of conservation of mass : “In all physical and chemical changes, the total mass of the reactantsis equal to that of the products” or “Matter can neither be created nor destroyed”.

Law of constant composition or definite proportions : “A chemical compound is always found to be made up of the same elements combined together in the same fixed ratio by weight”

Law of multiple proportions : “When two element combine together to form two or more chemicalcompounds, then the weights of one of the elements which combine with a fixed weight of the other bear asimple ratio to one another.”

Law of Reciprocal Proportions : The ratio of the weights of two elements A and B which combine with a fixed weight of the third element C is either the same or a simple multiple of the ratio of the weights of A and B which directly combine with each other.

Note : Law of reciprocal proportion can be used to obtain equivalent weights of elements.

Hence it is also called law of equivalent proportion.

Gay Lussac’s law of gaseous volumes : “When gases react together, they always do so in volumes which bear a simple ratio to one another and to the volumes of the products, if gaseous, all measurements aremade under the same conditions of temperature and pressure.

Dalton's Atomic Theory .It was based on laws of chemical combinations. The different assumptions of the theory are.

(1) All types of matter is composed of atoms.

(2) Atoms are extremely small, invisible and indestructible, i.e. atom can neither be destroyed nor created.

(3) Atoms of the same are alike in properties.

(4) Atoms combine in small whole numbers to form compound atoms (known as molecule).

(5) Compound atoms of a substance are alike posses similar properties.




Berzelius and Avogadro’s hypothesis.

Berzelius Hypothesis : Equal volumes of all gases contain equal number of atoms under same

conditions of temperature and pressure. When applied to law of combining volumes, this hypothesis predictsthat atoms are divisible and hence it is contrary to Dalton's atomic theory.

Avogadro’s hypothesis : “Equal volumes of all gases under similar conditions of temperature andpressure contain equal number of molecules.” Avogadro hypothesis is used in

(1) Deriving the molecular formula of gases. (2) Determining atomicity of gases.

(3) In deriving the relation, molecular wt. densityvapour 2 ×=

Atomic and Molecular mass.

Atomic Mass : It is the average relative mass of its atoms as compared with an atom of carbon –12

isotope taken as 12.

Atomic mass =12of atomanof Mass1/12

atomanof massAverage

C ×

Average atomic mass : If an elements exists in two isotopes having atomic masses ‘a’ and ‘b’ in the

ratio m : n, then average atomic mass = .)()(

nm

bnam

+×+×

Since the atomic mass is a ratio, it has no units and is expressed in amu, 1 amu = g 241066.1 −× .

Following are the important methods for determining atomic mass:

(a) Dulong and Petit's method (b) Specific heat method

(c) Volatile chloride formation method (d) Isomorphism method

Molecular mass : Molecular mass of a substance is the average relative mass of its molecules ascompared with an atom of C–12 isotope taken as 12. Thus, molecular mass is a ratio and hence has no units. Itis expressed in amu.

Molecular mass12of atomoneof Mass12/1

substanceof themoleculeoneof Mass

C ×=

Following are the important methods for determining molecular mass;

(a) Vapour density method, (b) Diffusion method, (c) Colligative properties and (d) Victor Mayer method

Difference between atomic mass and actual mass of an atoms.

The atomic mass of an atom (element) is not its actual mass. It is relative mass as compared with an atomof carbon taken as 12. It is expressed in amu. The actual mass of an atom means its mass in grams which isobtained by dividing the atomic mass of the element by Avogadro’s number (6.02×1023) because one gramatom contains Avogadro’s number of atoms.

Gram atomic & Molecular mass.

Gram atomic Mass (GAM) : The atomic mass of an element expressed in grams is called gram atomic

mass. This amount of the element is called one gram atom.




Gram molecular mass or molar mass (GMM) : The molecular mass of a substance expressed ingram is called gram molecular mass or molar mass. This amount of the substance is called one gram molecule.

(1) Aston, Dalton and avogram are the other names of 1 amu.

(2) Accurate determination of atomic weights and relative abundance of the isotopes can be determined with the help of an instrument known as mass spectrometer. Developed by Aston.

(3) In the beginning John Dalton determined the atomic weight with respect to hydrogen.

Mole concepts.

Mole is unit which represents 2310022.6 × particles (atoms molecules or ions, etc.) irrespective of their

nature. The number 2310022.6 × is called Avogadro's number and is represented by .0 N Avogadro's number

of gas molecules occupy a volume of 3400,22 cm at S.T.P. The number of molecules in 21cm of gas at STP is

called Loschmidt number. Its value if .10688.2 19× The calculation of this number is based on the Faraday'slaw of electrolysis.

Note : A mole any substance is related to mass of a substance, number of particles and volume

of a gaseous substance

Equivalent weights.

The equivalent weight of a substance is the number of parts by weight of the substance that combine withor displace directly or indirectly 1.008 parts by weight of hydrogen or 8 parts by weight of oxygen or 35.5 parts by weight of chlorine.

Eq. wt. of an element =Valency

massAtomic ; Eq. wt. of an acid =Basicity

acidwt.of theMol.

Eq. wt. of a base =Acidity

basetheof wt.Mol.; Eq. wt. of a salt =

charge-or chargeTotal

weightFormula

veve+

Eq. wt. of radicalchargeof unitsof No.

radicalof weightFormula=

Eq. wt. of an oxidizing or reducing agent =moleculesone bylostor gainedelectronsof Number

substancewt.of theMol.

Basicity is the number of displaceable H + ions from one molecule of the acid (e.g. 1 for HCl , 2 for,42SO H 3 for )43PO H .

Acidity is the number of displaceable OH – ions from one molecule of the base (e.g. 1 for NaOH , 2 forCa(OH )2 etc.

Following are the important methods of determining equivalent weight

(1) Hydrogen displacement method (for metals which can displace hydrogen from acids) –

Eq. wt. of metaldisplacedhydrogenof weight

metalof weight=

(2) Oxide method – Eq. wt. of metal 0.8oxygenof weight

metalof weight×=

(3) Chlorine method – Eq. wt. of metal 5.35chlorineof weight

metalof weight×=




Stoichometrys.

Calculation based on chemical equation is called stoichiometry. The chemical equation gives us important

information regarding –(1) Molar ratio between reactants and products. (2) Mass ratio between reactant and products.

(3) Volumes ratio between gaseous reactants and products.

The calculation based on chemical equation are of three types namely –(i) Mass-mass relationship, (ii) Mass-volume relationship, (iii) Volume-volume relationship

The various steps involved in solving problems based on chemical equations are.

(i) Write balanced equations for the reactions involved.(ii) Write the relative number of moles or relative masses of the reactants and products involved in the

problem below their formulae.(iii) In case of a gaseous substance, write down 22.4 L at STP below the formula in place of 1 mol (if

volume is involved).(iv) Apply unitary method to make the required calculations.

Emperical and molecular formulas.

Emperical formula - It is the simplest formula of a compound which gives the simplest whole numberratio of the atoms of the various elements present in one molecule of the compound e.g. emperical formula of

glucose .CHis)( 26126 OO H C

Calculation of emperical formulas

First calculate % of oxygen = 100 – Sum of % of all other elements. Then E.F. is calculated through thefollowing steps :

Element → % of element → Atomic mass → Relative no. of atoms =massAt.

%→ Simplest atomic ratio → Simplest whole no.

ratio

Molecular formula - It is the actual formula of a compound which gives the actual number of atoms of

various elements present in one molecule of the compound e.g. molecular formula of glucose is .6126 O H C

Relationship between emperical formula and molecular formula

Molecular formula = n× Emperical formula

Where n is any integer such as 1.,2,3…….etc.massformulaEmperical

massMolecules=n

Volumetric Analysis.(1) Normality of a solution is the number of gram equivalents of the solute present in one litre of the

solution.

(2) Molarity of a solution is the number of moles of the solute present in one litre of the solution.

(3) No. of gram equivalents =solutetheof wt.Eq.

gramsinsolutetheof Wt.

(4) No. of moles =substancetheof wt.Mol.

gramsinsubstancetheof Wt.

(5) When a solution is diluted, we can apply normality equation or molarity equation.

dilution)(Before

11 V N ×=

dilution)(After

22 V N ×or

dilution)Before(

11 V M ×=

dilution)(After

22 V M ×




(6) When the solutions of two substances ( an acid and a base or an oxidizing and a reducing substance)react completely, we apply normality equation viz.

1)solution(11 V N × =

2)Solution(22 V N ×

(7) In terms of molarity equation, for the reaction applyweProducts,21 →+ Bn An

)()(2

22

1

11

Bn

V M

An

V M =

Important formula used for numerical problems.

Mass of one atom of element =0 N

GAM . Mass of X atoms of element X

N

GAM ×=

0

.

Mass of 1 molecule of a substance0 N

GMM = . Mass of X molecule of a substance X N

GMM ×=0

.

No. of atoms in ‘w’ gram of element 0 N GMM

w×= . No. of atoms in ‘n’ moles of element 0 N n ×= .

No. of moles of element for X atom0 N

X = . No. of molecules in ‘w’ gm.of substance 0 N

GMM

w×= .

No. of molecules in ‘n’ moles of substance 0 N n ×= . No. of moles in X molecules0 N

X = .

Moles in ‘n’ atoms atomicity0

×= N

n . Eq wt. =Valency

ghtAtomic wei .

For elements with atomicity > 1, The formula can be modified as; Atoms in ‘n’ moles atomicity0 ××= N n .

Weight % of an element 100compoundof thewt.Mol.

molecule per atomsof No.wt.At.×

×= .

According to Dulong and Petit's rule for any element Atomic wt. × Specific heat = 6.4 (approx.) or

Approx. Atomic weight =heatSpecific

4.6

Atomic Structure

Structure of Atom.(1) John Dalton 1801, believed that matter is made up of extremely minute indivisible particles, called

atoms.

Atom consists of two parts

Nucleus : Neutron (n) +Proton ( p)

Extra nuclear part : Electron (e) only.

(2) The characteristics of fundamental particles are given below.

Fundamental particles




Particle

Symbol Mass inamu

Mass in Kg Charge in esu Charge in

coulomb

e/m Discovery

Electron 01e−

0.000548 31101091.9 −× ×− 803.4 1010

− 1910602.1 −×− g c /1076.1 8× Thomson

Proton 11 P +

1.00757 27106725.1 −× 1010803.4 −×+ 1910602.1 −×+ g c /1058.9 4× Goldstein(1886)

Neutron

10 n 1.00893 27106748.1

−× 0 0 0 J. Chadwick (1932)

Note : The mass of electron given above is the rest mass. It is 1/1837 times the mass of a hydrogen

atom.

(3) Other fundamental particles and Antiparticles

Bryons : These are heavy particles and include protons, neutrons etc.

Mesons : These are particles of intermediate mass.

Leptons : These are lighter particles viz. electron, particles and neutrions.

Positron : C.D. Anderson (1932) discovered it. It has same negligible mass and same amount of charge

as of the electron. But charge is+ve and of short life 810− sec.

Meson : Yukawa (1935) discovered it. It has mass in between that of proton and electron in cosmic rays.The charge may be +ve, –ve or zero.

Neutrino :)( Pauli (1927) suggested the existence of a particle called neutrino. It has a variable mass

less than that of an electron. The charge on this particle is zero. Allen and Rodebeek (1952), showed their freeexistence.

Antiproton ( − p ) : Segre (1956) discovered it. It has mass equal to that of a proton but with a negative

charge.

Allotropy : When the element exists in 2 or more forms in the same state that differ in physicalproperties, it is said to exhibit allotropy, while these forms are called allotropes of that element.

Note : The mass (m) of a particle (electron) at high speed is given by:

2

c-1

m rest

=

m , Where , = velocity of particle, =c velocity of light

(4) Rutherford's Experiment : In 1911, Rutherford observed that when − particles emitted from Ra

struck thin metallic sheets, many of them passed through the sheet with no change in their path but a few of

them got deflected o90 or through larger angles. He concluded that :

(i) As most of the α-particles passed undeflected, the atom must predominantly consists of empty space.

(ii) As a few − particles carrying +ve charge are strongly deflected there must be a heavy +ve charged body present in each atom and the volume occupied by this is only a minute fraction of the total volume of an

atom an atom. He called this +vely charged body as nucleus. It is surrounded by small negatively chargedparticles called electrons, at relatively large distances form the nucleus.

If ,c= then ∞=m .




In order to explain why the electrons do not fall into the nucleus due to electrostatic attraction,Rutherford proposed that electrons are revolving round the nucleus at high velocities. The centrifugal force

arising from this motion just balances the force of electrostatic attraction.Objections to Rutherford’s Model : When ever an electric charge is subjected to acceleration, it emits

radiation and loses energy. As a result of this, the orbit will become smaller and the electrons will drop on thenucleus. This, however, does not happen. This model also fails to explain the line spectrum of hydrogen.

(5) Atomic number and Mass number

Atomic number ( Z ) : Mosley denoted atomic no. of element by Z representing the no. of protons in thenucleus of an element.

= Z No. of protons in the nucleus of an atom = No. of electrons in the extra nuclear part of neutralatom.

Mass number (A) : It is equal to sum of numbers of protons and no. of neutrons in an atom.n p A += , or n Z A += , or n Z A =−

An element X having mass no. ( At . wt .) A and At . no. Z, is represented as A Z X

Note : Actual mass of atom is not equal to the sum of mass of e, p and n present in it, e.g., for

Chlorine )007276.1(17)( 3517 =Cl amu 17)008665.1(18 ++ amu (0.0005486) amu = 35.289005 amu. However,

the mass of chlorine has been accurately determined as 34.96885 amu. This difference between the two values(35.28901 amu – 34. 96885 amu)= 0.32016 amu is known as mass defect.

This difference, expressed in its energy equivalent, is called the binding energy of the nucleons (neutrons+ protons) in the nucleus of the atom in question.

(6) Different types of atomic species

Isotopes : Atoms of an element having the same atomic no. but different at. wt. are called isotopes.

eg Cl 3517 and ,37

17 Cl D H 21

11 , and ,3

1 T OO 178

168 , and O18

8

Isotopes have the same no. of protons and electrons but different no. of neutrons. They have the samechemical properties. The fractional at. wt. of an element is due to the different proportions in which various

isotopes are present in it, eg. chlorine has two isotopes 3517 Cl and 37

17 Cl present in the ratio 3:1.

Average at. wt. amu5.354

371353 =×+×=

Size of the nucleus = 1 Fermi = 10–15 mSize of the atom 1 Å = 10–10 m

Atom

Nucleus+

10–15 m

10–10

m

Unstability of atom

e–




Isobars : Atoms of different elements having different at. no. but the same at. wt. are called isobars. eg.

Ca K 4020

4019 ,

Isobars have different no. of protons, neutrons and electrons, but the sum of no. of protons and neutronsis same. They have different chemical properties.

Isotones : Atoms having the same no. of neutrons but different no. of protons are called Isotones. eg.

P Si 3115

3014 ,

Isoelectronic ions or molecules : Species having same no. of electron but different charge on nucleusare known as isoelectronic ions.

(a) ++++−−− 3,,,,, Al Mg Na Ne F O (b) −−−33 ,CO NO (c) ⊕O H NH 33 ,

(d) −CN CO N ,,2

(e) − NCS and 2

CS (f) +− Li He H ,,

Isodiaphers : Species with same isotopic number i.e. (A-2 Z ) or (n– Z )

(7) Electromagnetic radiations : “Electromagnetic radiations is a form of energy transported in theforms of waves from any source continuously with the velocity of light and consists of electric and magneticfields oscillating perpendicular to each other and to the direction of propagation”. Every wave has followingfive characteristics, viz,

Wavelength )( : The distance between two neighbouring troughs or crests. Expressed in cm, nm, Å

units. 1Å = .10101 81 cmnm −− =×

Frequency )( : The number of times a wave passes through a given point in one second. Expressed in

cps or Hz . 1 Hz = 1cps.

Velocity (c) : The distance travelled by the wave in one second. All waves have same velocity i.e.

.000,186103103 118110 −−− =×=× smiles sm scm

Wave number )( : The number of wavelengths per cm. Expressed in .1−cm

Amplitude )( : The height of the crest or depth of through of a wave. It determines the intensity or

brightness of the light beam.

Relation between wave characteristic :

cc==

(8) Electromagnetic spectrum : When sunlight is passedthrough a prism, it absorbs wavelength range of black colour radiationand other splits into a series of colour bands known as emissionspectrum and black colour band which is known as absorptionspectrum.

The splitting of light into seven colours is called emissionSpectrum.The characteristic range of wavelength of electromagneticradiation situated in an increasing or decreasing order calledelectromagnetic spectrum.

λ

VioletIndigoBlueGreen Yellow OrangeRed

PrismNarrow beam

The dispersion of light Screen




Wavelengths and frequencies of electromagnetic radiation’s

Electromagnetic radiation Wavelength(Å) Frequency (Hz or sec-1)

Radiowaves 13103 × to 9103 × 5101× to 9101 ×

Microwaves 9103 × to 6106 × 9101 × to 11105 ×

Infra red (IR) 6106 × to 7600 11105 × to 141095.3 ×

Visible 7600 to 3800 3. 95 1014 to 7.9 1014

Ultraviolet (UV) 3800 to 150 14109.7 × to 16102 ×

X–Rays 150 to 0.1 16102 × to 19103 ×

Gamma rays 0.1 to 0.01 19103 × to 20103 ×

Cosmic rays 0.01 to zero 20103 × to infinity

Limitations of electromagnetic waves : Electromagnetic wave theory was successful in explainingthe properties of light such as interference, diffraction etc. but it could not explain the phenomena of ‘Black body radiation’, and ‘Photoelectric effect.’

(9) Planck's quantum theory : This theory was proposed by Max Planck to explain black body radiations and photoelectric effect. According to this theory,

“Radiant energy is emitted or absorbed only in discrete units or packets of energy called photon

(quantum). the energy ' E ' associated with a quantum is given by h E = , where h is Planck's constant and v isfrequency of radiations”.

hchc

h E === =

1 Where h is Planck's constant =

.sec10625.6 34 J −× sec10625.6 27 erg −×=

c is velocity of light = 18 sec100.3 −× m = 110 sec100.3

−× cm

is frequency of light in ;sec 1− is wave no. in 1−m or .1−cm

It is thus clear that energy of photon decreases with increase in .

→ VIBGYOR increases and E decreases

Note : Energy ' E ' associated with a photon can also be written as

.12375

eV E

= where E is energy in eV and is wavelength of light in Å.

(10) Neil's Bohr Atomic Theory (1913) : it is based on the following assumptions.

(i) The electron moves around the +vely charged nucleus in a circular orbit, the centripetal force for thismotion is balanced by the electrostatic attraction.

(ii) The electron can rotate only in certain orbits which are known as stationary or quantized orbits for

which the angular momentum is an integral multiple of 2

h. When the electron moves in these orbits it cannot

emit any radiation.

Decreasing

λ

Increasingv,E




(iii) The electron radiates a quantum of energy of frequency v only when it jumps from an orbit of higher

energy to an orbit of low energy. If 1 E and 2 E are the energies of the electron in two successive orbits, then

hh E E (12 =− is Planck's constant= ergs271062.6 −× ).

c= ; =×= ,/103

10 scmc wavelength

The energy of the electron in an orbit characterised by quantum number 'n'

1

2

21

2

2

22

4228.13116.1320 −− =

−=−= KJmol

n

Z eVatm

n

Z

nh

me Z E n

Where, Z is atomic number; m and e are mass and charge of the electron respectively.

The radius of the paths in which an electron can revolve is given by;

Ånme

hnr 2

22

22

53.04

==

)101( 8 cm Å −=

Putting =n 1, 2, 3, 4 etc., the radii of the first five orbits of − H atom are:

Orbit (n) 1 2 3 4 5

Radius (Å) 0.529 2.116 4.761 8.464 13.225

When an electron jumps from an outer orbit in which its quantum number is '' 2n to an inner orbit in

which it is '' 1n ; the energy emitted as radiation is given by

−=−

22

21

2

422 11212 nnh

me Z E E nn

and the frequency expressed in wave number will be

−=

22

21

3

422 112

nnch

me Z

−=

22

21

11

nn R

Where R is the Rydberg constant. For hydrogen R is 109677.8 .1−cm

Bohr's model explains sucessfully the spectrum of hydrogen atom and species having one electron. Itexplains the appearance of five series of lines in hydrogen spectrum.

Comparative study of important spectral series of Hydrogen atom

S.No. Spectralseries

Lies in theregion

Transition

Rnn

nn

)( 21

22

22

21

max−

= R

n21

min = 21

22

22

min

max

nn

n

−=

1. Lymenseries

Ultravioletregion

11 =n

∞= ....4,3,22n

2and1 21 == nn

R3

4max =

∞== 21 and1 nn

R

1min =

34

min

max =




2. Balmerseries

Visibleregion

21 =n

∞= ....5,4,32n

3and2 21 == nn

R536max =

∞== 21 and2 nn

R4min = 5

9

min

max =

3. Paschenseries

Infra redregion

31 =n

∞= ....6,5,42n

4and3 21 == nn

R7

144max =

∞== 21 and3 nn

R

9min =

7

16

min

max =

4. Brackettseries

Infra redregion

41 =n

∞= ....7,6,52n

5and4 21 == nn

R9

2516max

×=

∞== 21 and4 nn

R

16min =

9

25

min

max =

5. Pfund

series

Infra red

region

51 =n

∞= ....8,7,62n

6and5 21 == nn

R11

3625max

×=

∞== 21 and5 nn

R

25min =

11

36

min

max

=

(11) Sommerfeld theory (1915) : It states that :

(i) The orbits in which the electrons move are in most cases elliptical.

(ii) The orbits are capable of interpenetrating and electrons in these orbits are spinning like a top.

(iii) The orbits are further divided into sub-orbits or sub-states which are denoted as ‘s’ ‘ p’ ‘d’ ‘ f’ orbitals.

A result of sommerfeld model suggests thataxisminor of length

axismajor of length

=k

n

e.g., if principal quantum no. .4=n

The values of k can be 1, 2, 3, 4 only, since k is an integer

∴ 4th shell have 4 subshells.

Total spin and magnetic moment : The total spin (s) of an atom is given by ,2

1n s ×= where n is

number of unpaired electrons.

The spin magnetic moment )( of electron (excluding orbital magnetic moment) in Bohr magneton

(B.M.) is given by : )1(4effective += s s

If n s ×= 2/1 ; ∴ ..)2(effective M Bnn +=

To derive the possible no. of in line spectrum, when an electron de-excites form one o level to other.

Say an electron is in 4th shell in H atom. It is to be de-excited to ground state level, i.e., I shell.

The possible no. of given out nΣ∆= 3)14( Σ=−Σ= 6321 =++=

(12) Photoelectric effect : The phenomenon of ejection of electrons from the surface of a metal whenlight of suitable frequency strikes on it, is called as photoelectric effect. The emitted electrons are calledphotoelectrons.

Threshold energy is a character of a particular metal, therefore




2

2

1 m xh +=

If the threshold frequency is ,0 then Threshold energy, ,0 h x = so that

20

2

1 mhh += or 0

2

2

1 hhm −=

Therefore kinetic energy of ejected electron )2

1(

2 m

0.. hh E K −=

In this equation = h energy of incidental photon and =0 h minimum value of energy required to eject

the electron.

(13) Particle and wave nature of electron i.e. dual nature : de Broglie proposed a relationship between of a moving particle with its velocity on the basis of quantum theory.

)(2 KE m

h

p

h

mu

h=== )

2

1(

2mu KE =

Where m is mass of moving particle, u is its velocity

P is momentum of particle equal to )(2or KE m , h is Planck's constant.

The circumference of the n th orbit (if closed) is equal to integer multiple of wavelength.

Thus, nr n =2

Also, Frequency of matter wave

u=

muh

u

/=

h

KE

h

mu 22

==

(14) Heisenberg's uncertainty principle : According to this principle, it is impossible to determinemomentum and position of a subatomic particle precisely and simultaneously.

4.

h x p ≥∆∆

4

.. h xum ≥∆∆

m

h xu

4. ≥∆∆

(15) Schrodinger wave equation : Bohr treated electron as a particle. However, de Broglie suggested

that −e has a dual nature, i.e., it behaves both as a particle as well as wave. The wavelength of movingparticle is

m

h= ....(1)

Where and m are the velocity and mass of moving particle respectively. If r is radius of the wave,

r 2 its circumference, then

Where p∆ is uncertainty in momentum

x∆ is uncertainty in position

u∆ is uncertainty in velocity.




r n 2= ....(2)

Thus according to wave theory, an electron is a stationary wave moving around the nucleus in a circular

path. The wave character was later on confirmed by Davison, Germer (1927) and Thomson (1928).

From eq. (1) and (2), we have

mv

nhr = 2 or

2

nhmvr =

According to Schrodinger, the electron does not move round the nucleus in fixed orbits, but may, infact, be any where with different probabilities. The probability of its presence near the nucleus is greatest and as thedistance from nucleus increases the probability decreases. Schrodinger from mathematical treatment of wavemotion gave a general wave equation describing the behaviour of a small particle. Consider a system such as astretched string. For its vibration,

.2

sin

x A=Ψ Where = x displacement , =Ψ wave function , = A amplitude of the wave, =

wavelength

0.).(8

2

2

2

2

=Ψ−+Ψ

E P E h

m

dx

d

2Ψ for an electron at a given point indicates the probability of occurrence of the electron at that point.

(16) The quantum numbers : The quantum numbers are results of Schrodinger wave equation.

(i) Principal quantum no.

• Denoted by 'n'

• The values of n are from 1 to n

1=n K shell; 2=n L shell

3=n M shell; 4=n N shell

• 'n' signify for the size and energy level of major energy shell.

(ii) Azimuthal or angular quantum no.

• Denoted by 'l '

•The values of l are from 0 to n – 1

0=l for s subshell; 1=l for p subshell

2=l for d subshell; 3=l for f subshell

• 'l ' signify for shape and energy level of subshells.

(iii) Magnetic quantum no.

• Denoted by 'm'




• The values of 'm' are from l ± to l ±

Let l =1 1−=m 0 +1

y x p p or z p y p or x p

Let 2=l 2−=m –1 0 +1

+2

xyd or xzd or 2 zd , yzd or 22 y x

d − ,

22 y xd − yzd xzd or xyd

• ‘m’ signify for the possible no. of orientations of subshells.

(iv) Spin quantum no

• Denoted by s

• The values of s are2

1+ or

2

1−

• ‘s’ Signify for the direction of spin of electron in a sub- subshell or orbital.

Angular momentum

Angular momentum of an electron in an orbital 2

hn=

Angular momentum of an electron in an orbital ])1[(2

l l h

+×=

(17) Pauli’s exclusion principle

(i) It is impossible for two electrons of an atom to have all their four quantum no. same.

(ii) e.g. is correct for 21 s

is wrong for 21 s

(iii) Following results have been obtained by Pauli’s exclusion principle.

• Maximum no. of electrons in a shell can be .2 2n

• Maximum no. of electrons in a subshell can be 14,10,6,2 in f d p s ,,, respectively.• Maximum no. of electrons in a sub-subshell (orbital) is 2 only.

(18) Rules for writing electronic configuration : The electronic configuration is written on the basisof following rules :

(i) Aufbau principle : The electrons in a poly electronic atom are filled one by one in order of increasingenergy level.

e.g. :1 H 11 s is correct, 12 s is wrong because energy level of .21 s s <

(ii) Hund's rules : In filling a group of orbitals of equal energy (or subshells) it is preferred to assignelectrons to empty orbitals rather than pair them in a particular subshell, because the former arrangementleads to lower energy level. Same spin of unpaired electrons in sub-subshell also gives rise to lower energy level.




e.g. :7 N 322 22,1 p s s

for 3

2 p is correctis wrong

is wrong

(iii) (n + l ) rule : The subshell with lower values of )( l n + possesses lower energy level and should be

filled first. e.g., :19 K 162622 333,22,1 d p s p s s is wrong

162622 4,33,22,1 s p s p s s is correct

l n + of 4044 =+= s

l n + of 5233 =+=d

Thus, 4s should be filled first.

If )( l n + is same for two subshells, the one with lower values of n possess lower energy and should be

filled first. e.g. :21 Sc 1262622 44,33,22,1 p s p s p s s is wrong

21626224,333,22,1 sd p s p s s is correct

l n + of 5144 =+= p

l n + of 5233 =+=d

Thus, d 3 should be filled first.

n of nd <3 of s4(iv) A subshell having nearly full filled or nearly half filled configuration tends to acquire exactly full filled

or exactly half filled nature in order to attain stability i.e. lower energy level.

e.g. :24 Cr 24626224,333,22,1 sd p s p s s is wrong

15626224,333,22,1 sd p s p s s is correct

:29 Cu 2962622 4,333,22,1 sd p s p s s is wrong

11062622 4,333,22,1 sd p s p s s is correct

:46 Pd 28621062622 5,444,333,22,1 sd p sd p s p s s is wrong

10621062622 444,333,22,1 d p sd p s p s s is correct

Number of nodes : Total number of nodes in a shell )1( −= n

Angular nodes l =

Spherical nodes 1−−= l n

***

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